Theorem hvadd23t 8842

Description: Commutative/associative law.

Assertion

Ref	Expression
hvadd23t	|- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A +h B) +h C) = ((A +h C) +h B))

Proof of Theorem hvadd23t

Step	Hyp	Ref	Expression
1		ax-hvcom 8810	. . . 4 |- ((B e. H~ /\ C e. H~) -> (B +h C) = (C +h B))
2	1	opreq2d 3967	. . 3 |- ((B e. H~ /\ C e. H~) -> (A +h (B +h C)) = (A +h (C +h B)))
3	2	3adant1 796	. 2 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> (A +h (B +h C)) = (A +h (C +h B)))
4		ax-hvass 8811	. 2 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A +h B) +h C) = (A +h (B +h C)))
5		ax-hvass 8811	. . 3 |- ((A e. H~ /\ C e. H~ /\ B e. H~) -> ((A +h C) +h B) = (A +h (C +h B)))
6	5	3com23 838	. 2 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A +h C) +h B) = (A +h (C +h B)))
7	3, 4, 6	3eqtr4d 1514	1 |- ((A e. H~ /\ B e. H~ /\ C e. H~) -> ((A +h B) +h C) = ((A +h C) +h B))

Syntax hints:   -> wi 3   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956  (class class class)co 3954  H~chil 8727   +h cva 8728

This theorem is referenced by:  hvadd4t 8844  hvadd23 8860

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-hvcom 8810  ax-hvass 8811

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fv 3193  df-opr 3956

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